Formulae for the Distribution of Ions under an Ideal Mask

Analytical expressions are derived for the distribution of implanted ions under an ideal mask using Pearson II, Pearson VII, and modified Gaussian models for the lateral distribution of a single ion. These expressions are then used to calculate ion concentration contours under an ideal mask from fits to single-ion Monte Carlo generated projected range profiles and depth-dependent lateral standard deviations and kurtoses for 275 keV boron and phosphorus ions implanted into amorphous silicon. These contours are then compared with ion concentration contours under an ideal mask obtained from numerical convolution of the raw, two-dimensional Monte Carlo profile data with the aperture function. Good agreement is found over three orders of magnitude.